# Engineering Tables/Properties of Derivatives

Derivative Conditions
1 ${\displaystyle {\frac {d}{dx}}c=0}$
2 ${\displaystyle {\frac {d}{dx}}(cx)=c}$
3
Elementary Power Rule
${\displaystyle {\frac {d}{dx}}(x^{n})=nx^{n-1}}$
4
Sum Rule
${\displaystyle {\frac {d}{dx}}\left(f\pm g\pm h\pm \cdots \right)={\frac {df}{dx}}\pm {\frac {dg}{dx}}\pm {\frac {dh}{dx}}\pm \cdots }$
5
Constant Multiple Rule
${\displaystyle {\frac {d}{dx}}(cf)=c{\frac {df}{dx}}}$
6
Product Rule
${\displaystyle {\frac {d}{dx}}(fg)=f{\frac {dg}{dx}}+g{\frac {df}{dx}}}$
6
Product Rule (Extended)
${\displaystyle {\frac {d}{dx}}(fgh)=fg{\frac {dh}{dx}}+fh{\frac {dg}{dx}}+gh{\frac {df}{dx}}}$
7
Quotient Rule
${\displaystyle {\frac {d}{dx}}\left({\frac {f}{g}}\right)={\frac {g{\frac {df}{dx}}-f{\frac {dg}{dx}}}{g^{2}}}}$ ${\displaystyle g\neq 0\,}$
8
Chain Rule
${\displaystyle {\frac {df}{dx}}={\frac {df}{dg}}\cdot {\frac {dg}{dx}}}$
9 ${\displaystyle {\frac {d}{dx}}\left(f^{n}\right)=nf^{n-1}{\frac {df}{dx}}}$
10
Reciprocal Rule
${\displaystyle {\frac {d}{dx}}\left({\frac {1}{f}}\right)=-{\frac {1}{f^{2}}}{\frac {df}{dx}}}$ ${\displaystyle f\neq 0\,}$
11
Functional Power Rule
${\displaystyle {\frac {d}{dx}}\left(f^{g}\right)={\frac {d}{dx}}\left(e^{g\ln f}\right)=f^{g}\left({\frac {g}{f}}\cdot {\frac {df}{dx}}+{\frac {dg}{dx}}\ln f\right)}$ ${\displaystyle f>0\,}$
12
Logarithm Rule
${\displaystyle {\frac {df}{dx}}=f{\frac {d}{dx}}\left(\ln f\right)}$ ${\displaystyle f>0\,}$
13
Inverse Function Rule
${\displaystyle {\frac {df}{dx}}={\frac {1}{\frac {dx}{df}}}}$ ${\displaystyle {\frac {dx}{df}}\neq 0\,}$
Notes:
1. f, g, h are functions of x
2. c, n are constants.